Abstract
In some applied problems of signal processing, the maximum of a sample of \(\chi ^2(m)\) random variables is computed and compared with a threshold to assess certain properties. It is well known that this maximum, conveniently normalized, converges in law to a Gumbel random variable; however, numerical and simulation studies show that the norming constants that are usually suggested are inaccurate for moderate or even large sample sizes. In this paper, we propose, for Gamma laws (in particular, for a \(\chi ^2(m)\) law) and other Weibull-like distributions, other norming constants computed with the asymptotics of the Lambert \(W\) function that significantly improve the accuracy of the approximation to the Gumbel law.
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Acknowledgments
The first author was partially supported by grants MINECO reference MTM 2013-40998-P and Generalitat de Catalunya reference 2014-SGR568. The second author by the European Space Agency (ESA) through the DINGPOS contract AO/1-5328/06/NL/GLC, and by the Spanish Government and Generalitat de Catalunya through grants TEC2011-28219 and 2014-SGR1586, respectively. The third author by grants MINECO reference MTM2012-33937 and Generalitat de Catalunya reference 2014-SGR422. The authors thank two anonymous referees for their careful reading of the manuscript and their suggestions that helped to improve the paper.
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Gasull, A., López-Salcedo, J.A. & Utzet, F. Maxima of Gamma random variables and other Weibull-like distributions and the Lambert \(\varvec{W}\) function. TEST 24, 714–733 (2015). https://doi.org/10.1007/s11749-015-0431-9
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DOI: https://doi.org/10.1007/s11749-015-0431-9